@Michael – re your first question, we're assuming here that \\(a, b, c, d\\) are all mutually distinct (there's no "collapsing" going on).

If you set \\(a = I\\) then you can prove that \\(c\otimes d \leq c\\) and \\(c\otimes d \leq d\\).

But that means \\(c\otimes d\\) is a common lower bound of \\(c\\) and \\(d\\) – and no such lower bound exists.

Intuitively I think the best way of looking at this is that the partial order is too irregular to admit a monoidal structure: \\(a\\) and \\(b\\) don't have a join, \\(c\\) and \\(d\\) don't have a meet, there is no top element, nor is there a bottom one... basically \\(\leq\\) is too badly behaved for any \\(\otimes\\) operation to respect it.

If you set \\(a = I\\) then you can prove that \\(c\otimes d \leq c\\) and \\(c\otimes d \leq d\\).

But that means \\(c\otimes d\\) is a common lower bound of \\(c\\) and \\(d\\) – and no such lower bound exists.

Intuitively I think the best way of looking at this is that the partial order is too irregular to admit a monoidal structure: \\(a\\) and \\(b\\) don't have a join, \\(c\\) and \\(d\\) don't have a meet, there is no top element, nor is there a bottom one... basically \\(\leq\\) is too badly behaved for any \\(\otimes\\) operation to respect it.